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| NAME
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allocatedNumber of labeled simple graphs covering n vertices with at least one forendpoint Gusor Wisemanleaf.
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| DATA
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0, 0, 1, 3, 31, 515, 15381, 834491, 83016613, 15330074139, 5324658838645, 3522941267488973, 4489497643961740521, 11119309286377621015089, 53893949089393110881259181, 513788884660608277842596504415, 9669175277199248753133328740702449
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| OFFSET
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0,4
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| COMMENTS
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Covering means there are no isolated vertices.
A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.
Also graphs with minimum vertex-degree 1.
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| FORMULA
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Inverse binomial transform of A245797, if we assume A245797(0) = 0.
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| EXAMPLE
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The a(4) = 31 edge-sets:
{12,34} {12,13,14} {12,13,14,23}
{13,24} {12,13,24} {12,13,14,24}
{14,23} {12,13,34} {12,13,14,34}
{12,14,23} {12,13,23,24}
{12,14,34} {12,13,23,34}
{12,23,24} {12,14,23,24}
{12,23,34} {12,14,24,34}
{12,24,34} {12,23,24,34}
{13,14,23} {13,14,23,34}
{13,14,24} {13,14,24,34}
{13,23,24} {13,23,24,34}
{13,23,34} {14,23,24,34}
{13,24,34}
{14,23,24}
{14,23,34}
{14,24,34}
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| MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]], {n, 0, 5}]
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| CROSSREFS
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The non-covering version is A245797.
The generalization to set-systems is A327229.
BII-numbers of set-systems with minimum degree 1 are A327105.
Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327228, A327230.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Sep 01 2019
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| STATUS
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approved
editing
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